|
| |
|
|
A007863
|
|
Number of hybrid binary trees with n nodes.
|
|
12
| |
|
|
1, 2, 7, 31, 154, 820, 4575, 26398, 156233, 943174, 5785416, 35955297, 225914342, 1432705496, 9158708775, 58954911423, 381806076426, 2485972170888, 16263884777805, 106858957537838
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
REFERENCES
| N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
J.M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50, 1994, 135-145.
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to rooted trees
|
|
|
FORMULA
| G.f. satisfies x^2*A(x)^3+x*A(x)^2+(-1+x)*A(x)+1 = 0.
a(n) = 3F2({-n, n+1, n+2 } ; {1, 3/2})( -(1/4) ) [From Olivier GERARD, Apr 23 2009]
a(n) = 1/(n+1)*sum(k=0..n, binomial(n+k,n)*binomial(n+k+1,n-k)). [From Vladimir Kruchinin, Dec 24 2010]
G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*A(x)^k] * x^n/n ). [Paul D. Hanna, Feb 13 2011]
|
|
|
MAPLE
| A:= proc(n) option remember; if n=0 then 1 else convert (series ((x^2 *A(n-1)^3 +x*A(n-1)^2 +1)/ (1-x), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n): seq (a(n), n=0..19); # Alois P. Heinz, Aug 22 2008
|
|
|
MATHEMATICA
| InverseSeries[Series[(y-y^2-y^3)/(1+y), {y, 0, 24}], x] (* then A(x)=y(x)/x *) - Len Smiley Apr 14 2000
Table[ HypergeometricPFQ[{-n, 1 + n, 2 + n}, {1, 3/2}, -(1/4)], {n, 0, 20}] [From Olivier GERARD, Apr 23 2009]
|
|
|
PROG
| (Macsyma) taylor_solve_choose_order:true; taylor_solve( A^3*X^2+A^2*X+A*(X-1)+1, A, X, 0, [ 20 ]);
(PARI) {a(n)=sum(k=0, n, binomial(n+k, n)*binomial(n+k+1, n-k))/(n+1)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*(A+A^2)+x^2*A^3); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}
|
|
|
CROSSREFS
| Cf. A007788.
Sequence in context: A076177 A007164 A126033 * A030823 A030873 A030913
Adjacent sequences: A007860 A007861 A007862 * A007864 A007865 A007866
|
|
|
KEYWORD
| nonn,changed
|
|
|
AUTHOR
| pallo(AT)u-bourgogne.fr (Jean Pallo)
|
| |
|
|
